Abstract
Let \mathcal R be a finite set of integers satisfying appropriate local conditions. We show the existence of long clusters of primes p in bounded length intervals with p-b squarefree for all b \in \mathcal R . Moreover, we can enforce that the primes p in our cluster satisfy any one of the following conditions: (1) p lies in a short interval [N, N+N^{{7}/{12}+\epsilon}] , (2) p belongs to a given inhomogeneous Beatty sequence, (3) with c \in ({8}/{9},1) fixed, p^c lies in a prescribed interval mod 1 of length p^{-1+c+\epsilon} .
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